shear wall 10

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Finite element formulation for lateral torsional buckling analysis of shear deformable mono-symmetric thin-walled members Arash Sahraei a , Liping Wu b , Magdi Mohareb a,n a Department of Civil Engineering, University of Ottawa, Ottawa, ON, Canada K1N 6N5 b Candu Energy Inc., 2285 Speakman Drive, Mississauga, ON, Canada L5K 1B1 article info Article history: Received 26 April 2014 Received in revised form 5 October 2014 Accepted 25 November 2014 Available online 23 January 2015 Keywords: Thin-walled members Finite element Mono-symmetric sections Shear deformable members Lateral torsional buckling abstract A shear deformable theory and a computationally efcient nite element are developed to determine the lateral torsional buckling capacity of beams with mono-symmetric I-sections under general loading. A closed-form solution is also derived for the case of a mono-symmetric simply supported beam under uniform bending moments. The nite element is then used to provide solutions for simply supported beams, cantilevers, and developing moment gradient factors for the case of linear moments. The formulation is shown to successfully capture interaction effects between axial loads and bending moments as well as the load height position effect. The validity of the element is veried through comparisons with other established numerical solutions. & 2014 Elsevier Ltd. All rights reserved. 1. Motivation Wide ange mono-symmetric sections are commonly used as girders in bridge structures. In buildings, they represent a viable design alternative as exural members in cases such as roof members where positive bending moments induced by gravity load combinations involving gravity loads can be signicantly larger than negative moments typically induced by wind uplift. When such members are used in large span laterally unsupported beams, their resistance is frequently governed by lateral torsional buckling resistance. Relatively recently, design standards (e.g., CAN/CSA-S16-09 [1], ANSI/AISC 360-05 [2] and the subsequent edition ANSI/AISC 360-10 [3]) have incorporated provisions for quantifying the lateral torsional buckling resistance for simply supported mono-symmetric members under general loading. More complex cases involving continuous beams, cantilever sus- pended constructions, cantilevers, are beyond the scope of North American design standards, although, as will be discussed in the literature review, past research has tackled some of these issues. The present study contributes to the existing body of knowledge by developing a theory and nite element for the buckling analysis of mono-symmetric sections. In a recent study, Wu and Mohareb [4,5] developed a shear deformable theory and nite element formulation for lateral torsional buckling of thin-walled members. The theory was limited to members with doubly symmetric sections, and the resulting element exhibited slow convergence characteristics, thus requiring a several hundreds of degrees of freedom to model simple problems. Within this context, the present study is intended to advance the work in [4,5] in two respects; (a) it extends the developments to beams with mono- symmetric sections, and (b) it devises an effective interpolation scheme to accelerate the convergence characteristics of resulting nite element. 2. Literature review The present work is concerned with the lateral torsional buckling of beams of mono-symmetric sections based a shear deformable thin-walled theory. Thus, within the vast body of research about lateral torsional buckling, the present review focuses on the work related to beams of mono-symmetric cross- sections (Section 2.1) and recent buckling solutions under shear deformable theories (Section 2.2). 2.1. Lateral torsional buckling for members of mono-symmetric cross-sections Several studies have investigated the lateral torsional buckling resistance of mono-symmetric I-beams. Using the nite integral method, Anderson and Trahair [6] developed tables for the critical loads of cantilevers and simply supported beams. Based on energy Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/tws Thin-Walled Structures http://dx.doi.org/10.1016/j.tws.2014.11.023 0263-8231/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. E-mail address: [email protected] (M. Mohareb). Thin-Walled Structures 89 (2015) 212226

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Page 1: shear wall 10

Finite element formulation for lateral torsional buckling analysisof shear deformable mono-symmetric thin-walled members

Arash Sahraei a, Liping Wu b, Magdi Mohareb a,n

a Department of Civil Engineering, University of Ottawa, Ottawa, ON, Canada K1N 6N5b Candu Energy Inc., 2285 Speakman Drive, Mississauga, ON, Canada L5K 1B1

a r t i c l e i n f o

Article history:Received 26 April 2014Received in revised form5 October 2014Accepted 25 November 2014Available online 23 January 2015

Keywords:Thin-walled membersFinite elementMono-symmetric sectionsShear deformable membersLateral torsional buckling

a b s t r a c t

A shear deformable theory and a computationally efficient finite element are developed to determine thelateral torsional buckling capacity of beams with mono-symmetric I-sections under general loading.A closed-form solution is also derived for the case of a mono-symmetric simply supported beam underuniform bending moments. The finite element is then used to provide solutions for simply supportedbeams, cantilevers, and developing moment gradient factors for the case of linear moments. Theformulation is shown to successfully capture interaction effects between axial loads and bendingmoments as well as the load height position effect. The validity of the element is verified throughcomparisons with other established numerical solutions.

& 2014 Elsevier Ltd. All rights reserved.

1. Motivation

Wide flange mono-symmetric sections are commonly used asgirders in bridge structures. In buildings, they represent a viabledesign alternative as flexural members in cases such as roofmembers where positive bending moments induced by gravityload combinations involving gravity loads can be significantlylarger than negative moments typically induced by wind uplift.When such members are used in large span laterally unsupportedbeams, their resistance is frequently governed by lateral torsionalbuckling resistance. Relatively recently, design standards (e.g.,CAN/CSA-S16-09 [1], ANSI/AISC 360-05 [2] and the subsequentedition ANSI/AISC 360-10 [3]) have incorporated provisions forquantifying the lateral torsional buckling resistance for simplysupported mono-symmetric members under general loading.More complex cases involving continuous beams, cantilever sus-pended constructions, cantilevers, are beyond the scope of NorthAmerican design standards, although, as will be discussed in theliterature review, past research has tackled some of these issues.The present study contributes to the existing body of knowledgeby developing a theory and finite element for the buckling analysisof mono-symmetric sections. In a recent study, Wu and Mohareb[4,5] developed a shear deformable theory and finite elementformulation for lateral torsional buckling of thin-walled members.

The theory was limited to members with doubly symmetricsections, and the resulting element exhibited slow convergencecharacteristics, thus requiring a several hundreds of degrees offreedom to model simple problems. Within this context, thepresent study is intended to advance the work in [4,5] in tworespects; (a) it extends the developments to beams with mono-symmetric sections, and (b) it devises an effective interpolationscheme to accelerate the convergence characteristics of resultingfinite element.

2. Literature review

The present work is concerned with the lateral torsionalbuckling of beams of mono-symmetric sections based a sheardeformable thin-walled theory. Thus, within the vast body ofresearch about lateral torsional buckling, the present reviewfocuses on the work related to beams of mono-symmetric cross-sections (Section 2.1) and recent buckling solutions under sheardeformable theories (Section 2.2).

2.1. Lateral torsional buckling for members of mono-symmetriccross-sections

Several studies have investigated the lateral torsional bucklingresistance of mono-symmetric I-beams. Using the finite integralmethod, Anderson and Trahair [6] developed tables for the criticalloads of cantilevers and simply supported beams. Based on energy

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/tws

Thin-Walled Structures

http://dx.doi.org/10.1016/j.tws.2014.11.0230263-8231/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: [email protected] (M. Mohareb).

Thin-Walled Structures 89 (2015) 212–226

Page 2: shear wall 10

solutions, Robert and Burt [7] developed a lateral torsional solutionfor beams with boundary conditions similar to those reported in [6].Both studies [6,7] focused on members under concentrated anduniformly distributed loads. Using the Raleigh Ritz method, Wangand Kitipornchai [8] developed buckling solutions for cantilevers andsimply supported beams [9] under concentrated and uniformlydistributed loads. Also, Kitipornchai et al. [10] investigated the effectof moment gradient on the buckling resistance of simply supportedbeams. Based on the stationarity condition of the total potentialenergy, Kitipornchai and Wang [11] investigated the elastic lateraltorsional buckling resistance of the simply supported tee beamsunder moment gradient. They showed that for inverted tee beams,uniform bending moment is not the most severe loading case and thecases involving high moment gradients ordinarily were more critical.Using shell finite element analysis, Helwig et al. [12] modelled thelateral buckling capacity of girders subject to transverse point anduniformly distributed loads. Attard [13] developed solutions forestimating elastic lateral torsional capacity of beams with mono-symmetric and doubly symmetric sections and general boundaryconditions. Using Ritz and Galerkin’s methods, Mohri et al. [14]developed an analytical model for estimating the lateral torsionalbuckling resistance of simply supported beams under concentratedand uniformly distributed loads. Andrade et al. [15] extended theapplication of three-factor lateral torsional buckling formula in the

Eurocode [16] to mono-symmetric cantilevers subject to uni-formly distributed and concentrated transverse tip loads applied.Their solution incorporated the effect of load height. Based on theprinciple of stationarity of the second variation of the totalpotential energy, Zhang and Tong [17] developed a new theoryfor estimating the lateral torsional buckling capacity of cantile-vers subject to concentrated and uniformly distributed loads anduniform bending moments. Mohri et al. [18] developed linear andnonlinear models to investigate into the lateral torsional bucklingcapacity of simply beams under moment gradient. Using ahyperelastic constitutive model, Attard and Kim [19] formulatedlateral torsional buckling solutions for shear deformable simplysupported beams subject to uniform bending moment. Using theGeneralized Beam Theory (GBT), Camotim et al. [20] modeledbeams with fork-type end supports under uniform moment, mid-span point load, two-point loads, distributed load and linearmoments. They observed that among all loading conditionsincluding end moments and transverse loads applied at shearcenter, the lowest critical buckling moments do not necessarilycorrespond to uniform bending moment. Mohri et al. [21]developed a non-linear model to investigate the effect of axialforces on lateral torsional buckling resistance of simply sup-ported I and H-sections. Their solutions involved concentratedand uniformly distributed loads.

List of symbols

ai to zi elements of stiffness matricesai to f i elements of stiffness matricesAh i

; B� �

; Ch i

matrices which are coefficients of quadratic eigen-value problem

A cross-sectional areaAp a poleAi i¼ 1 to 8ð Þ integration constantsBðzÞ½ � matrix relating displacement fields to integration

constantsC section centroidDhh;Dxh;Dxx;Dyk;

Dyyk;Dyh;Dyωh;

Dxx0y0 ;Dωx0y0

properties of cross-section related to shear

deformationdðzÞ� �T

field displacementsE modulus of elasticityE zð Þ½ � diagonal matrix of exponential functionsG shear modulusH½ � matrix relating nodal displacements to integration

constantsI½ � identity matrixIxx; Iyy moments of inertia of the cross-section about x-axis

and y-axis respectivelyIpx polar moment of inertia about x-axisIωω warping constantJ St. Venant torsional constantKf� �

stiffness matrix due to flexural stressesKG½ �N geometric matrix due to normal forcesKG½ �M geometric matrix due to bending momentsKG½ �V geometric matrix due to shear forcesKG½ �qy geometric matrix due to load position effect of the

distributed transverse loadKG½ �qz geometric matrix due to load position effect of the

distributed axial loadKG½ �Qy geometric matrix due to load position effect of the

concentrated transverse load

Ks½ � stiffness matrix due to shear stressesKsv½ � stiffness matrix due to Saint Venant shear stressl length of a finite elementL span of the memberL zð Þ½ � matrix of shape functionsmi roots of quadratic eigenvalue problemM1;M2 internal bending moment at both ends of an elementMxpðzÞ resultant of the moments of the normal stresses

obtained from pre-buckling analysisN1;N2 internal normal forces at both ends of an elementNpðzÞ resultant of the normal stresses obtained from pre-

buckling analysisO origin of the Cartesian coordinates x, y and zqy; qz distributed load applied to a member acting along the

y- and z-direction, respectivelyS0 sectorial originSC shear center of the cross-sectionub lateral buckling displacementuNh iT vector of nodal displacementsU internal strain energyV potential energyV1;V2 internal shearing forces at both ends of an elementVypðzÞ resultant of shear force component along y-direction

obtained from pre-buckling analysisx; y; z Cartesian coordinatesyA coordinate of the shear center along y-directionα constantβ end moment ratioλ load multiplierπ total potential energyδπ variation of total potential energyθyb;θzb Buckling rotation angles about y, z axes, respectivelyΦ½ � matrix of eigenvectorsσzz normal stress along z directionτzs shear stress on the cross-section mid-surfaceωðsÞ warping function or sectorial area of a cross-sectionψ b warping deformation (1/length)

A. Sahraei et al. / Thin-Walled Structures 89 (2015) 212–226 213

Page 3: shear wall 10

Table 1Comparative studies on lateral torsional buckling of mono-symmetric I-beams.

Author(s) Boundary conditiontypes

Loading types Effects captured Solutions developed Analysis type

Simplysupported

Cantilever Concentratedtransverse load(s)

Uniformlydistributed load

Uniformbendingmoment

Linearmoment

Axialload

Distortionaleffects

Sheardeforma-tions

Pre-bucklingdeformations

Closed-form

FEA Other numericalmethods

Anderson andTrahair [6]

✓ ✓ ✓ ✓ Finite integral

Bradford [29] ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Roberts andBurt [7]

✓ ✓ ✓ ✓ ✓ ✓ Stationarity of the totalpotential energy

Kitipornchaiet al. [10]

✓ ✓ ✓ Rayleigh–Ritz

Wang andKitipornchai[9]

✓ ✓ ✓ Rayleigh–Ritz

Wang andKitipornchai[8]

✓ ✓ ✓ Rayleigh–Ritz

Attard [13] ✓ ✓ ✓ ✓ ✓ ✓ ✓ Stationarity of the totalpotential energy

Helwig et al.[12]

✓ ✓ ✓ Shell FEA

Mohri et al. [14] ✓ ✓ ✓ Ritz and Galerkin

Andrade et al.[15]

✓ ✓ ✓ Rayleigh–Ritz

Zhang and Tong[17]

✓ ✓ ✓ ✓ Stationarity of the totalpotential energy

Erkmen andMohareb [40]

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Stationarity of thecomplementary energy

Mohri et al. [18] ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Beam 3D FEA

Attard and Kim[19]

✓ ✓ ✓ ✓ Hyperelastic

Camotim et al.[20]

✓ ✓ ✓ ✓ ✓ Stationarity of the totalpotential energy

LTBEAM,Shell FEA,and GBT

Mohri et al. [21] ✓ ✓ ✓ ✓ ✓ ✓ Galerkin Beam 3D FEA

Erkmen [30] ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Hellinger–ReissnerPresent study ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ Stationarity of the total

potential energy

A.Sahraei

etal./

Thin-Walled

Structures89

(2015)212

–226214

Page 4: shear wall 10

In addition to the above solutions, several finite element formula-tions have been developed for the lateral torsional buckling of mono-symmetric sections. This includes the work of Krajcinovic [22] andBarsoum and Gallagher [23] who developed a finite element forbuckling analysis based on the Vlasov thin-walled beam theory [24].Based on the principle of stationarity of the second variation of thetotal potential energy, Attard [25] developed two finite elementformulations for estimating lateral torsional buckling loads of beams.Papangelis et al. [26] developed a computer program to predictelastic lateral torsional buckling estimates of beams, beam-columnsand plane frames. Distortional effects in doubly and mono-symmetric sections were also investigated in the work of Hancocket al. [27], Bradford and Trahair [28] and Bradford [29]. Using theHellinger–Reissner principle, Erkmen [30] developed a hybrid finiteelement formulation for shear deformable elements. Lateral torsionalbuckling solutions for web-tapered mono-symmetric beams wereinvestigated in [31–36]. Also, solutions for laminated compositeinclude the work in [37,38]. Table 1 provides a comparative summaryof the most relevant studies. As shown in the table, the present studyaims at developing a general theory and finite element formulationfor the lateral torsional buckling analysis of mono-symmetric mem-bers. The solution captures warping and shear deformation effectsand excludes pre-buckling and distortional effects. It is applicable togeneral boundary and loading conditions and incorporates thedestabilizing effects of axial loading, shearing force, and bendingmoments.

2.2. Buckling solutions under shear deformable theories

Other shear deformable theories were also developed. Thisincludes the work of Erkmen and Mohareb [39] who developed acomplementary energy variational principle and formulated a finiteelement (Erkmen and Mohareb [40]) for thin-walled members withopen cross-sections. In a subsequent study, focused on torsionalbuckling of columns, Erkmen et al. [41] demonstrated that the elastictorsional buckling of columns can be guaranteed to converge frombelow.

3. Assumptions

The following assumptions have been adopted

1. The formulation is restricted to prismatic thin-walled memberswith mono-symmetric sections consisting of segments parallelto the principal axes,

2. Regarding shear/bending action, the cross-section remainsrigid in its own plane during deformation but does not remainperpendicular to the neutral axis after deformation in line withthe Timoshenko theory. The hypothesis is further generalizedfor torsion/warping action. Similar kinematics have been usedin buckling problems in Saade [42], Kollar [43] Back and Will[44], Attard and Kim [19], Kim and Lee [37] and Lee [38], andWu and Mohareb [4,5].

3. The material is assumed to be linearly elastic and obeys theHooke’s law,

4. Strains are assumed small but rotations are assumed to bemoderate. Rotation effects are thus included in the formulationby retaining the non-linear strain components,

5. The member buckles in an inextensional mode [45] whichmeans that throughout buckling, the centroidal longitudinalstrain and curvature in yz-plane remain zero. This signifies thatthe member is assumed to buckle under constant axial load andbending moments, and

6. The solution neglects pre-buckling deformation effects.

4. Variational formulation

This section outlines the details of the variational formulation.The treatment is similar to that presented in Wu and Mohareb [4]for doubly symmetric sections. As such, only important milestonesare provided here and reader is referred to Wu and Mohareb [4]for more a more thorough discussion of the methodology. A right-handed Cartesian coordinate system is adopted in which the Z-axisis oriented along the axial direction of the member while X-axisand Y-axis are parallel to major and minor principal axes of thecross-section, respectively. The origin is taken to coincide with thecross-section centroid Cðxc ¼ 0; yc ¼ 0Þ while pole Ap is taken tocoincide with the shear center SC xA ¼ 0; yA

� �.

4.1. Problem description and notation

The member is assumed to be subjected to a uniformlydistributed transverse load qy applied at a distance yqyðzÞ fromthe shear center and a uniformly distributed axial load qz acting atdistance yqzðzÞ from the origin. Under such external loads, themember deforms from configuration 1 to 2 as shown in Fig. 1 andundergoes displacements vp zð Þ, wp zð Þ and rotation θxp zð Þ. As aconvention, subscript p represents pre-buckling displacement,strain, and stress fields. The applied loads are assumed to increaseby a factor λ and attain the values λqy and λqz at the onset ofbuckling (Configuration 3). Under the load increase, it is assumedthat pre-buckling deformations linearly increase to λvp zð Þ, λwp zð Þand λθp zð Þ. The section then undergoes lateral torsional buckling(Configuration 4) manifested by lateral displacement ub, weak-axisrotation θyb, angle of twist θzb and warping deformation ψ b. Again,as a matter of convention, subscript b denotes field displacements,strains, or stresses, occurring during the buckling stage (i.e., ingoing from configuration 3 to 4) while superscripts n denotes thetotal fields (i.e., in going from configuration 1 to 4).

4.2. Kinematic relations

Under the kinematic assumptions postulated above, a point Son the section mid-surface can be shown [4] to undergo totaldisplacements un

s s; zð Þ; vns s; zð Þ; wns s; zð Þ given by

un

s s; zð Þ ¼ ub zð Þ� y sð Þ�yA� �

θzb zð Þ

vns s; zð Þ ¼ λvp zð Þþx sð Þθzb zð Þ

wn

s s; zð Þ ¼ λwp zð Þþy sð Þλθxp zð Þ�x sð Þθyb zð Þþω sð Þψ b zð Þ

þx sð Þλθxp zð Þθzb zð Þþy sð Þθyb zð Þθzb zð Þ ð1a–cÞ

Fig. 2 depicts the global coordinate system, displacements, andsign conventions adopted in this study.

Fig. 1. Different stages of deformation.

A. Sahraei et al. / Thin-Walled Structures 89 (2015) 212–226 215

Page 5: shear wall 10

4.3. Conditions of neutral stability

The condition of neutral stability is given by evoking thevariation of the second variation of the total potential energy π,[4], i.e.,

δ12δ2π

� ¼ δ

12δ2Uþδ2V ��

¼ 0 ð2Þ

in which, U is the internal strain energy and V is the load potentialenergy gained by externally applied loads. The variation of theirsecond variation of U and V are given by

δ 12 δ2U �h i

¼ δ 12

R L0

RA E δεzzb

� �2þλσzzpδ2εzzb

h innþ G δγzsb

�2þλγzspδ

2τzsb

� �dAdzþ1

2

Z L

0GJ δθ0

zb

� �2dz:

δ12δ2V ��

¼ δ12δ2Uþδ2V ��

�Z L

0�12qy yqy�yA �

δθzb� �2�

þqz yqzδθybδθzb

�idzg ð3a–bÞ

Eqs. (1a–c) are differentiated with respect the appropriatecoordinates to yield the strain expressions. The first variationand second variations of strains are

δεzzb ¼ �xδθ0ybþωδψ 0

b

δγzsb ¼dxds

δu0b�δθyb

� �þh δθ0zbþδψ b

� �δ2εzzb ¼ δu0

b

� �2�2 y�yA� �

δu0bδθ

0zbþ x2þ y�yA

� �2h iδθ0

zb

� �2þ2y δθ0

ybδzbþδθybθ0zb

�δ2γzsb ¼ 2

dyds

δθybδθzb� �þx

dxds

δθybδθ0yb

��xh δψbδθ

0yb

��

�ωdxds

δθybδψ 0b

� �þωh δψ bδψ0b

� � � dyds

� dkds

� δu0

bδθzb� �

þkdkds

� δθ0

zbδθzb� � ð4a–dÞ

where, h sð Þ ¼ x sð Þ sin α sð Þ� y sð Þ�yA� �

cos α sð Þ, k sð Þ ¼ x sð Þ cos α sð Þþy sð Þ�yA� �

sinα sð Þ. From Eqs. (4a–d) by substituting into the vibra-

tional expression δ 1=2δ2π �

¼ δ 1=2 δ2Uþδ2V �h i

¼ 0, and recal-

ling the pre-buckling stress expressionsσzzp ¼Np zð Þ=AþMxp zð Þy=Ixx and τzzp ¼ Gγzsp , one obtains

δ12δ2π

� ¼ δ

12δ2Ubþδ2Usvþδ2Usþδ2VNþδ2VMþδ2VV

þδ2Vqyþδ2Vqz

� ¼ 0 ð5Þ

in which, Ub is the internal strain energy due to normal stresses, Usv isthe internal strain energy due to Saint–Venant shear stress, Us is theinternal strain energy due to shear stresses, VN is the destabilizingterm of the total potential energy due to normal forces, VM is thedestabilizing term of the total potential energy due to bendingmoments, VV is the destabilizing term of the total potential energydue to shear forces, Vqyis the destabilizing term of the total potentialenergy due to transverse load position effect, Vqz is the destabilizingterm of the total potential energy due to longitudinal load positioneffect. Under this condition, the second variations of the above energyterms take the form

δ2Ub ¼Z L

0EIyy δθ0

yb

�2þEIωω δψ 0

b

� �2� dz

δ2Usv ¼Z L

0GJ δθ0

zb

� �2dz

δ2Us ¼ GZ L

0Dxxδu0

b�DxxδθybþDxhδθ0zbþDxhδψ b

� �δu0

b

�þ �Dxxδu0

bþDxxδθyb�Dxhδθ0zb�Dxhδψ b

� �δθyb

þ Dxhδu0b�DxhδθybþDhhδθ

0zbþDhhδψ b

� �δθ0

zb

þ Dxhδu0b�DxhδθybþDhhδθ

0zbþDhhδψ b

� �δψ b

�dz

δ2VN ¼ �λZ L

0

Np zð ÞA

� �Au0

b2� 2yAA� �

u0bθ

0zb

�� Ixxþ IyyþyA

2A� �

θ0zb2�dz

δ2VM ¼ �λZ L

0

MxpðzÞIxx

� �2Ixxðθ0

ybθzbþθybθ0zbÞþ2Ixxu0

bθ0zb

h

� Ipx�2yAIxx� �

θ0zb

2idz

δ2VV ¼ �λZ L

0

2VypðzÞDyy

� �DyyθybθzbþDyyku

0bθzb�Dykθ

0zbθzb

��Dxx0y0θybθ

0ybþDωx0y0θybψ 0

bþDyhψ bθ0yb�Dyωhδψ bδψ

0b

�dz

δ2Vqy ¼ �λZ L

0�qy yqy�yA

�δθzb� �2h i

dz

δ2Vqz ¼ �λZ L

02qzyqzδθybδθzbdz ð6a–hÞ

in which, the following sectional properties have been defined.

Iyy ¼RAx

2dA Ixy ¼RAxydA Iωω ¼ RAω2dA Ipx ¼

RAy x2þy2� �

dA

Sx ¼ZAydA Sxω ¼

ZAxωdA Dhh ¼

ZAh2dA Dxx ¼

ZA

dxds

� 2

dA

Dxh ¼ZAh

dxds

� dA Dyy ¼

ZA

dyds

� 2

dA Dyk ¼ZAkdyds

dkds

dA

Dyyk ¼ZA

dyds

� 2dkds

dA

Dyh ¼ZAxh

dyds

� dA Dyωh ¼

ZAhω

dyds

dA Dxx0y0 ¼ZAxdxds

dyds

dA

Dωx0y0 ¼ZAωdxds

dyds

dA

It is observed that for common cross-sections consistingexclusively of segments parallel to x and y axes (such as a mono-symmetric I section), all area integral terms containing the productdy=ds� �

dx=ds� �

vanish, i.e., Dxx0y0 ¼Dωx0y0 ¼ 0.As a verification of the validity of Eqs. (6a–h), when the cross-

section is doubly symmetric constants Ixy; Ipx; Sx; Sxω; Dxh; Dyk;

Dyh and Dyωh vanish and the present variational statement revert tothat in Wu and Mohareb [4] when the coordinate system is taken tobe orthogonal.

Fig. 2. Global coordinate system and displacement components.

A. Sahraei et al. / Thin-Walled Structures 89 (2015) 212–226216

Page 6: shear wall 10

4.4. Finite element formulation I

The variational expressions in Eqs (6a–h) consist of the bucklingdisplacement functions ub θyb θzb ψ b and their first derivativeswith respect to coordinate z. Thus, each of the assumed fourdisplacement functions needs to satisfy only C0 continuity. By takingtwo nodes per element, and adopting a linear interpolation schemebetween the two nodal values, the displacement fields ub zð Þ θyb zð Þ�θzb zð Þψ b zð ÞiT are related to the nodal displacements through

ub zð Þ θyb zð Þ θzb zð Þ ψ b zð ÞD E

1�4

¼ Hb zð Þ� �1�2

u1

u2

( )2�1

θy1

θy2

( )2�1

θz1

θz2

( )2�1

ψ1

ψ2

( )2�1

24

35

ð7a–dÞin which, Hb zð Þ� �

1�2 ¼ 1�z=L� �

z=L� �D E

is the vector of shapelinear shape functions andu1;u2;θy1; ::::ψ2 are the nodal displace-ment. In a similar manner, the pre-buckling stress resultantsNp zð Þ Vyp zð Þ Mxp zð Þ are linearly interpolated between the internalforces N1;N2;V1;V2;M1;M2 at the nodes as obtained from the pre-buckling analysis (Fig. 3b, d, f), i.e.,

Np zð Þ Vyp zð Þ Mxp zð ÞD E

¼ Hb zð Þ� �1�2

N1

�N2

( )2�1

V1

�V2

( )2�1

M1

�M2

( )2�1

24

35ð8a–cÞ

The resulting element is similar to that reported in [5] (and willbe subsequently referred to as the WM element), with twodifferences. The present element is geared towards mono-symmetric sections while the WM element is for doubly sym-metric sections. Also, the present formulation is based on anorthogonal coordinate system while the WM is based on generalnon-orthogonal coordinates.

4.5. Finite element formulation II

The element developed in Section 4.4 has a minimal number ofdegrees of freedom (8 DOFs for the buckling solution) but will beshown to exhibit slow convergence characteristics, thus needinghundreds of elements to solve simple problems. Within this context,the present section aims at developing an element which preservesthe low number degrees of freedom per element while accelerating itsconvergence characteristics. This is achieved by adopting differentinterpolation schemes for pre-buckling internal forces and bucklingdisplacement fields.

4.5.1. Approximation of pre-buckling internal forcesIn general, the pre-buckling internal forces Np zð Þ, Vyp zð Þ and

Mxp zð Þ are non-constant functions. Under such conditions, theclosed-form solution of the governing neutral stability conditionsstemming from the above variational principle become unattain-able. Thus, the non-constant internal forces obtained from pre-buckling analysis (Fig. 3a, c, e) are approximated as piecewiseconstant functions equal to the average values of the internalforces. Thus, one can set

Np zð Þ �Np ¼N1�N2ð Þ

2;Vyp zð Þ � Vyp ¼

V1�V2ð Þ2

;Mxp zð Þ �Mxp

¼ M1�M2ð Þ2

ð9a–cÞ

in the variational statement (Eqs. (6a–h)) within the subdomain Le(Fig. 3b, d, f) of the element. When the number of elements issufficiently large, the piecewise representation of pre-buckling internalforces would approach that of actual internal force distributions andthe resulting approximate total potential energy expression of thesystemwill approach that based on Eqs. (6a–h). This treatment will beshown rather advantageous from a computational viewpoint and willlead to desirable convergence characteristics.

Fig. 3. Internal forces for a beam-column: (a) normal forces within member, (b) idealized constant normal force within the element, (c) shearing forces within member,(d) idealized constant shearing force within the element, (e) bending moments within member, and (f) idealized bending moment within the element.

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4.5.2. Approximate equations of neutral stabilityUnder the approximations introduced in Section 4.5.1, Eqs. (6a–h)

are integrated by parts and like terms are grouped together. Notingthat the functions δub, δθyb, δθzb and δψ b are arbitrary, one recoversthe conditions of neutral stability

in which, D¼ d=dz¼ d=Ledξ is the first derivative of displacementfields with respect to the dimensionless coordinate ξ¼ z=Le.

4.5.3. Formulating shape functionsIt is proposed to generate shape functions which satisfy Eq. (10).

The presence of the unknowns λNp; λVyp; λMxp makes such a solutionunattainable given that λ is unknown a-priori. As such, the termsinvolving λNp; λVyp; λMxp are assumed negligible. This assumptionturns out to be accurate for beams of practical dimensions.

Another issue arising when solving Eq. (10), is the need toestimate. λqyðyqy�yAÞ=GDxx When the load λqy is a applied at theshear center, i.e., yqy ¼ yA, it is clear that λqyðyqy�yAÞ=GDxx wouldvanish. Otherwise, the order of magnitude for maximum forthe distributed load λqy can be estimated by equating thelateral torsional buckling resistance of the beam Cbðπ2EIyy=2L2Þ

βxþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ2x þ4 GJL2=π2EIyyþ Iωω=Iyy

�r� , in which, Cb ¼ 1:14, to the

external moments λqyL2=8 for a simply supported beam, and solving

for λqy. It is clear that the proposed scheme will yield approximatevalue for λqy since it does not necessarily capture the end cond-itions of the element, yielding approximate shape functions. Never-theless, the approximate functions thus obtainedwill be shown to havesuperior convergence characteristics compared to that in formulation I.

4.5.4. Closed-form solution for the field equationsThe coupled system of equations (Eq. (10)) has constant coeffi-

cients. Its solution is assumed to take the form ub=Le;θyb;�

θzb; Leψ biT ¼ Ai;Bi;Ci;Di� �Temiz . By substituting into Eq. (10), one

obtains the quadratic Eigen value problem

m2i Ah i

þmi B� �þ C

h i �4�4

φi

� �4�1 ¼ 0f g4�1 ð11Þ

in which, matrices Ah i

, B� �

and Ch i

are defined this time indimensional form as

Ah i

¼

�GDxx 0 �GDxh 00 EIyy 0 0

�GDxh 0 �G JþDhhð Þ 00 0 0 EIωω

26664

37775 ð12Þ

B� �¼

0 GDxx 0 �GDxh

GDxx 0 GDxh 00 GDxh 0 �GDhh

�GDxh 0 �GDhh 0

266664

377775 ð13Þ

Ch i

¼

0 0 0 00 �GDxx 0 GDxh

0 0 λqy yqy�yA �

0

0 GDxh 0 �GDhh

266664

377775 ð14Þ

in which, φi is the eigenvector corresponding the eigenvalue mi.Eq. (11) can be expressed into the following equivalent lineareigenvalue problem

B� �

Ch i

� I½ � 0½ �8�8

24

35þmi

Ah i

0½ �0½ � I½ �

24

358�8

0@

1A mi φi

� �φi

� �( )8�1

¼ 00

� �8�1

ð15Þin which, I½ � ¼Diag 1 1 1 1

� �T is the identity matrix. Theabove eigenvalue problem is observed to have four zero roots,

ð10Þ

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i.e., m1 ¼m2 ¼m3 ¼m4 ¼ 0. Thus, the closed-form solution takesthe form

ub ¼ A1þA2zþA3z2þA4z3þA5em5zþA6em6zþA7em7zþA8em8z

θyb ¼ B1þB2zþB3z2þB4z3þB5em5zþB6em6zþB7em7zþB8em8z

θzb ¼ C1þC2zþC3z2þC4z3þC5em5zþC6em6zþC7em7zþC8em8z

ψ b ¼D1þD2zþD3z2þD4z3þD5em5zþD6em6zþD7em7zþD8em8z

ð16a–dÞThe remaining four roots mi (i¼5 to 8) can either be obtained

by solving the right eigenvalue problem in Eq. (15) or from theequivalent bi-quadratic characteristic equation

G �D2xhE

2GIωωIyyþDhhDxxE2GIωωIyyþDxxE

2GIωωIyyJ �

m4nþ ED2

xhG2IyyJ�EDhhDxxG

2IyyJh

þDxxE2IωωIyyqyyA

im2

þEGIyyqyyA D2xh�DhhDxx

�o¼ 0 ð17Þ

It is observed that for the special case of doubly symmetricsections, one has Dxh ¼ yA ¼ 0 and the last coefficient of Eq. (17)vanishes. In such a case, one obtains six repeated zero roots,and Eq. (16a–d) becomes an invalid solution. Thus, as stated inAssumption 1, the present solution is restricted to mono-symmetric sections. By substituting Eq. (16a–d) and their deriva-tives into the field equations (Eq. (10)), and performing algebraicsimplifications, the 32 integration constants Ai �Di ði¼ 1:::8Þcan be reduced to eight independent constants Ai ði¼ 1:::8Þ.The field displacements dðzÞ� �T ¼ ub zð Þ θyb zð Þ θzb zð Þ ψ b zð Þ

D ETare thus related to integration constants A

� �T ¼A1 A2 A3 A4 A5 A6 A7 A8� �T through

d zð Þ� �¼ BðzÞ½ �4�8 Af g8�1 ð18Þwhere,

BðzÞ½ �4�8 ¼ F zð Þ½ �4�4 Φ½ �4�4 E zð Þ½ �4�4

h i; F zð Þ½ �4�4

¼

1 z z2 z3

0 1 2z 3z2� 6EIyyDhh

G Dxh2 �DhhDxxð Þ

0 0 0 00 0 0 �6EIyyDxh

G Dxh2 �DhhDxxð Þ

2666664

3777775 ð19a–bÞ

and Φ½ �4�4 ¼ ϕ� �

1 ϕ� �

2 ϕ� �

3 ϕ� �

4

h iis the matrix of eigen-

vectors of the quadratic eigenvalue problem defined in Eq. (15) andE zð Þ½ �4�4 ¼Diag em5z; em6z; em7z; em8z½ � is the diagonal matrix of expo-nential functions. The Integration constants Ai can be related to thenodal displacements uNh iT ¼ ub1 θyb1 θzb1 ψ b1 u

�b2θyb2θzb2ψ b2iT by setting z¼ 0 and z¼ L in Eq. (18) leading to

uNf g8�1 ¼ H½ �8�8 Af g8�1 ð20Þin which,

H½ �8�8 ¼F 0ð Þ½ �4�4 Φ½ �4�4 E 0ð Þ½ �4�4

F lð Þ� �4�4 Φ½ �4�4 E lð Þ� �

4�4

" #ð21Þ

From Eq. (20) by solving for the integration constants andsubstituting into Eq. (18), one obtains

d zð Þ� �¼ L zð Þ½ �4�8 uNf g8�1 ð22Þin which, L zð Þ½ �4�8 ¼ B zð Þ½ �4�8 H½ ��1

8�8 is a matrix of shape function, and

ub zð Þ ¼ L1ðzÞ� �T

1�8 uNf g8�1 ¼ p1� �T

1�4 L zð Þ½ �4�8 uNf g8�1

θyb zð Þ ¼ L2ðzÞ� �T

1�8 uNf g8�1 ¼ p2� �T

1�4 L zð Þ½ �4�8 uNf g8�1

θzb zð Þ ¼ L3ðzÞ� �T

1�8 uNf g8�1 ¼ p3� �T

1�4 L zð Þ½ �4�8 uNf g8�1

ψ b zð Þ ¼ L4ðzÞ� �T

1�8 uNf g8�1 ¼ p4� �T

1�4 L zð Þ½ �4�8 uNf g8�1 ð23a–dÞ

and, p1� �T

1�4 ¼ 1 0 0 0� �

1�4, p2� �T

1�4 ¼ 0 1 0 0� �

1�4,p3� �T

1�4 ¼ 0 0 1 0� �

1�4, and p4� �T

1�4 ¼ 0 0 0 1� �

1�4 havebeen defined. It is noted that when a section is doubly symmetric,matrix H½ �8�8 becomes singular and the shape functions introduced inEq. (22) become unattainable. From Eqs. (23a–d), by substituting intoEqs. (6a–h) and then Eq. (5), one obtains

δun� �T K½ �f þ K½ �svþ K½ �sþλ KG½ �Nþ KG½ �Mþ KG½ �V þ KG½ �qy

��þ KG½ �qz

�g δun� �¼ 0 ð24Þ

in which, K½ �f is the elastic stiffness matrix due to flexural stresses,K½ �sv is the elastic stiffness matrix due to Saint Venant shear stresses,K½ �s is the elastic matrix due to the remaining shear stresses, KG½ �N isthe geometric matrix due to normal forces, KG½ �M is the geometricmatrix due to bending moments, KG½ �V is the geometric matrix due toshear forces, KG½ �qy is the geometric matrix due to the distributedtransverse load and KG½ �qz is the geometric matrix due to thedistributed axial load. These stiffness matrices are obtained from

Kf� �

; Ksv½ �; Ks½ �; KG½ �N ; KG½ �M ; KG½ �V ; KG½ �qy; KG½ �qz� �¼

H�1h iT

8�8M1½ �; M2½ �; M3½ �; M4½ �; M5½ �; M6½ �; M7½ �; M8½ �� �

H½ ��18�8

ð25Þin which, M1½ � to M8½ � are provided in Appendix A.

5. Examples

This section provides various buckling examples aimed at asses-sing the quality of the results, and illustrate its various features. Allexamples assume steel material with E¼ 200;000 MPa andG¼ 77;000 MPa and all the examples (excluding Example 7), arerelated to the section illustrated in Fig. 4. Cross-sectional proper-ties are Ixx ¼ 5:6987� 107 mm4, Iyy ¼ 1:42155� 107 mm4, A¼8� 103 mm2, Iωω ¼ 3:2080� 1010 mm6, Ipx ¼ 2:1056� 109 mm5,yA ¼ �58 mm, J ¼ 8:61867� 105 mm4, Dxx ¼ 5:600� 103 mm2,Dhh ¼ 5:71264� 107 mm4, Dyy ¼ 2:400� 103 mm2, Dxh ¼�2:52800� 105 mm3, and Dyk ¼ �72;000 mm3.

5.1. Example 1: Closed-form solution for a simply supported beamunder uniform bending moment

A simply supported beam of length L with a mono-symmetriccross-section is subject to uniform bending moment Mx. All otherinternal forces are assumed to vanish. By setting Mxp zð Þ ¼�Mxpa0;Nz ¼ Vy ¼ qy ¼ qz ¼ 0Þ in the governing differential equa-tions (Eq. (10)) and solving the resulting coupled system of

Fig. 4. Dimensions of the mono-symmetric cross-section.

A. Sahraei et al. / Thin-Walled Structures 89 (2015) 212–226 219

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differential equations, one obtains

d zð Þ� �¼ BðzÞ½ �4�8 Af g8�1 ð26Þ

The relevant boundary conditions are

ub 0ð Þ ¼ 0; EIyyθ0yb 0ð ÞþMxpθzb 0ð Þ ¼ 0; θzb 0ð Þ ¼ 0; ψ 0

b 0ð Þ ¼ 0

ub Lð Þ ¼ 0; EIyyθ0yb Lð ÞþMxpθzb Lð Þ ¼ 0; θzb Lð Þ ¼ 0; ψ 0

b Lð Þ ¼ 0

ð27a–hÞ

From Eq. (26) by substituting into the displacement fieldequations in Eqs. (27a–h), one obtains

Mcr ¼ λMxp1;2 ¼ �b7ffiffid

p2a

a¼ �E2IyyIωωπL

�4�GE DhhIyyþDxxIωω

� � πL

�2þG2 D2

xh�DhhDxx

��

b¼ EGIyyπL

�2EIωω 2DxhþβxDxx

� � πL

�2�Gβx D2

xh�DhhDxx

��

c¼ EG2IyyπL

�2EIωω �D2

xhþDhhDxxþDxxJ � π

L

�2�GJ D2

xh�DhhDxx

��

d¼ b2�4ac ð28a–eÞ

In Eq. (28a–e), it can be verified that by setting Dxh ¼ βx ¼ 0,one recovers the critical moment expression Mcrd based the sheardeformable theory as provided in Wu and Mohareb [4], i.e.,

Mcrd ¼ 7Gπ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDhhDxxE

2IωωIyyπ2þDxxE2IωωIyyJπ2þEDhhDxxGIyyJL

2

DhhDxxG2L4þE2IωωIyyπ4þEDhhGIyyπ2L2þEDxxGIωωπ2L2

vuut ð29Þ

also, for a large spans L, one has �bπLð Þ2EIyy

� �2β2

x

� �=2a

πLð Þ2EIyy2β2

x

� 2�e

( )-0,b

ffiffiffie

p=2a

πLð Þ2EIyy2β2

x

� 2�e

( )-0,

ffiffiffid2a

pπ=Lð Þ2EIyy

� �2β2

x

� �=

πLð Þ2EIyy2β2

x

� 2�e

( )-0,�

ffiffiffiffied2a

q=

πLð Þ2EIyy2β2

x

� 2�e

( )-1,in which,

e¼ π=L� �2EIyyh i

=2n o2

β2x þ4 GJL2=π2EIyy

�þ Iωω=Iyy� �h in o

and

one can show that the critical moment expression in Eq. (28a)approaches that of the classical solution [45], i.e.,

Mcl ¼π2EIyy2L2

βx7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ2x þ4

GJL2

π2EIyyþ Iωω

Iyy

!vuut24

35 ð30a–bÞ

In Eqs. (28a) and (30), the positive sign is taken when the largeflange is in compression.

5.2. Example 2: Mesh density analysis

Consider a 2 m span cantilever section subject to a verticalconcentrated load located at the tip and acting at the shear center.The critical load as determined by the present formulation isprovided for various discretizations. Results based on the presentformulation (i.e., based on formulation II are compared to thosepredicted by (a) the WM element, (b) the classical non-sheardeformable element by Barsoum and Gallagher [23] (referred to asBG) and (c) the WM element (Table 2). Also, for comparison, asolution was performed under the ABAQUS S4R shell element.ABAQUS S4R solution yields the lowest buckling load estimate of282.3 kN. This is due to the fact that the shell element capturesboth the distortional and shear deformation effects and thusprovides the most flexible representation of all solutions. Thepresent element and the WM element predict nearly equalbuckling loads of 287.2 kN and 288.0 kN, respectively. These valuesare slightly higher than that based on the shell solution. Whileshear deformation is captured in both formulations, they do notaccount for distortional effect and thus they provide a slightlystiffer representation for the member. The largest buckling load isthat based on the BG which predicts a buckling load of 329.5 kN.This is expected since the BG element neglects shear distortionaland shear deformation effects and thus provides the stiffestrepresentation of the member among all solutions. Under thepresent formulation, it is observed that no more than eightelements are needed to attain convergence, in a manner similarto the BG element, but in contrast with the significantly largernumber of WM elements needed.

The present solution converges from below for the problem, i.e., acoarser mesh tends to under-predict the buckling loads. This con-trasts to the WM and BG elements which consistently converge fromabove. However, convergence from below cannot be guaranteed. Thisis illustrated by the considering a 5 m span simply supported beamunder end reverse moments (Table 3). For the case of a singleelement, the approximation Mxp zð Þ �Mxp ¼ M1�M2ð Þ=2 introducedin Eq. (9c) yields, Mxp ¼ 0, thus vanishing the destabilizing term dueto bending moment, and the only destabilizing term remaining is thedue to shear (which is minor in the present 5 m span beam), yieldinga high buckling moment prediction of 286,800 kNm. A significantpredictive improvement is obtained by taking eight elements.

Table 2Mesh density study for cantilever under a concentrated load at the tip (span¼2 m, ABAQUS critical load¼282.3 kN).

Present study WM element BG element

Number ofelements

Buckling load(kN)

Present study/ABAQUS (%)

Number ofelements

Buckling load(kN)

WM/ABAQUS(%)

Number ofelements

Buckling load(kN)

BG/ABAQUS(%)

2 275.9 97.73 32 302.7 107.2 2 332.2 117.73 283.6 100.4 64 291.4 103.2 3 330.0 116.94 285.7 101.2 128 288.4 102.2 4 329.7 116.85 286.5 101.5 256 288.0 102.0 5 329.6 116.86 286.9 101.6 – – – 6 329.6 116.88 287.2 101.7 – – – 8 329.5 116.7

10 287.2 101.7 – – – – – –

Table 3Convergence study for a simply supported beam(span¼5 m) under reverse end moments.

Number of elements Buckling moment (kNm)

1 286:8� 103

2 602.44 529.98 499.9

10 498.1

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5.3. Example 3: Influence of span on shear deformation effects

A cantilever is subject to a transverse concentrated load appliedat the shear center of the free end. Two spans are examined in thisexample; 1000 and 4000 mm. The lateral torsional buckling load isestimated based on four solutions: (1) The classical BG solutionwhich neglects shear deformations and distortional effects, (2) theWM solution [5] which captures shear deformations but neglectsdistortional effects, (3) the present formulation which also capturesshear deformations and neglects distortional effects and (4) theABAQUS shell analysis which considers both effects. As suggested inExample 2, eight elements were used for the BG solution, 64elements per meter were used for the WM element, and eightelements were used for the present solution. In the ABAQUS model,10 elements per top flange, four elements per bottom flange, 10elements along the web height and 50 elements in the longitudinaldirection were taken to model the beam. The results are presentedin Table 4. As observed in Example 2, the ABAQUS shell elementsolution provides a lowest buckling prediction. This is attributed tothe fact that the shell formulation is the only solution that capturesdistortional effects, which tend to be more significant in a short spancantilever. This is illustrated in Fig. 5a where the web of the 1 mspan cantilever is observed to undergo minor distortion near the topflange (relative to the shown straight reference line). In contrast, theweb for the 4 m span (Fig. 5b) cantilever is observed to essentiallyundergo no distortion compared to the straight reference line. Sinceboth the WM element and the present element capture sheardeformation effects, their buckling load predictions are smaller thanthose based on the BG element. As illustrated by results, the sheardeformation effect is more pronounced in the short span cantilever.This is evident by the 24% difference observed between the bucklingload prediction based on the present element and the BG element.The difference is only 6% for the longer span cantilever.

5.4. Example 4: Beam under linear bending moment

A simply supported mono-symmetric beam is subject to alinear bending moment distribution as shown in Fig. 6a. A strongaxis moment Mx is a applied at the left end and a moment varyβMx at the other end where �1rβr1. Two cases are consid-ered: In Case 1, the larger flange is located in the top of the sectionso that the larger flange is under compression (Fig. 6b) and inCase 2, the smaller flange is in the top so that the smaller flange isunder compression (Fig. 6c). Spans were taken to vary from 1 m to5 m. For Case (a) under uniform moments, i.e., β¼ �1, the lateraltorsional buckling capacity is predicted based on three differentFEA solutions including present element (with eight elementsalong the span), and the WM and BG elements, the closed-form asprovided in Eq. (28a–e) as well as the classical solution as givenby Eq. (30a–b) while retaining the positive sign. Table 5 showsthat the critical moment as predicted by the present FEA andclosed form solution agree well with the classical solution forlarge spans (2 m and larger). For the 1 m span, where sheardeformation effects are significant, the critical moment as pre-dicted by the present shear deformable solution is less than thatpredicted by the classical solution which neglects shear deforma-tion effects. It is noted that the results based present elementnearly coincide the WM element with a significantly lowernumber of DOFs.

For non-uniform moments, the moment gradient factor isdefined as the ratio of the critical moment as predicted by thepresent study to that of classical solution as given by Eq. (30a).This ratio accounts for the end moment ratios β and the span. Theresults are depicted in Fig. 7 for Case 1 where moments inducecompression in the larger flange and Fig. 8 for Case 2 where thesmaller flange is under compression. For the longer spans, themoment gradient factor is observed to be almost identical for 3 m,4 m and 5 m spans (and larger spans –not shown on the figure) inthe moment gradient range �1rβr0 i.e., when the larger flangeis under entirely under compression. For shorter spans, smallermoment gradient factors are obtained given that shear deforma-tion effects gain significance in such short spans. When momentsinduce compression in the smaller flange, the moment gradientfactor monotonically increases with the end moment ration β(Fig. 8). In contrast, for the case where the larger flange is undercompression, the moment gradient factors peak around β¼ 0:5for 3 m, 4 m, 5 m spans and close toβ¼ 0:0 for the shortspan beams.

Table 4Buckling loads (kN) for a mono-symmetric cantilever beam under a tip verticalconcentrated load.

Span(mm)

ABAQUS Presentelement

WMelement

BGelement

Present/ABAQUS

WM/ABAQUS

BG/ABAQUS

1000 866.5 925.5 930.8 1223 1.07 1.07 1.414000 81.86 84.50 84.90 89.70 1.03 1.04 1.10

Fig. 5. Distorted cross-section at free end: (a) L¼1000 mm, (b) L¼4000 mm.

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Fig. 6. Simply supported beam under moment gradient (a) elevation, (b) cross-section for Case (1)—moments Mx induces compression in larger flange, and (c) cross-sectionfor Case (2)—moments Mx induces compression in smaller flange.

Table 5Lateral torsional buckling loads (kN m) for a simply supported beam under uniform bending moment.

Span (m) 1 2 3 4 5

Present finite element (8 elements) 4652 1441 767.8 508.4 376.4WM element (64 elements/m) 4658 1444 768.9 509.0 376.6BG element (8 elements) 5017 1466 773.0 510.0 376.9Closed-form solution present study—(Eq. 28a) 4647 1440 764.5 508.2 376.0Classical closed-form solution Mcl—(Eq. 30a) 5017 1466 772.9 510.0 376.9Present/classical 0.93 0.98 0.99 1.00 1.00

Fig. 7. Moment gradient factor versus various end moment ratios β and spans (m)—for Case (1): larger flange under compression.

Fig. 8. Moment gradient factor versus various end moment ratios β and spans (m) —for Case (2): smaller flange under compression.

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5.5. Example 5: Axial force–bending interaction

A simply supported member has a 4 m span and is subject to anaxial compressive force Qz and two equal end moments. In theabsence of uniform bending moment, the flexural–torsional bucklingload Qz0 obtained is 1580.6 kN while from Example 4, the bucklingmoment Mx0 in the absence of axial load is 508.4 kNm. In order todevelop the Qz�Mx buckling interaction diagram, several loadcombinations Qzi;Mxið Þ i¼ 1; :::;n are applied to the member andthe buckling eigenvalues are obtained for each case. This givesi¼ 1; :::;n critical load combinationsλi Qzi=Qz0;λiMxi=Mx0

� �in which

each load combination has been normalized with respect to Qz0 andMx0. The resulting normalized interaction curve is depicted in Fig. 9.

As can be seen, unlike doubly symmetric sections, the diagramis non-symmetric about the horizontal axis. This observation is inline with what is observed in [21]. The higher critical momentratio λMx=Mx0 ¼ 1 is obtained when the section is under purebending and when the top flange is under compression.

5.6. Example 6: Effect of load height position for a member underconcentrated transverse load

A cantilever spanning 5 m is subject to a concentrated trans-verse load applied at the tip. Three different load positions areconsidered: (a) top flange, (b) shear center and (c) bottom flange.The results are shown to agree well with those based on the shellfinite element analysis (Table 6). Due to the destabilizing effect of

top flange loading, buckling loads are lower than that based onshear center loading. Bottom flange loading is associated with astabilizing effect which increases the buckling load.

5.7. Example 7: Mono-symmetric I-girder

The present example illustrates the applicability of the formulationfor other types of mono-symmetric sections. A simply supported girder(cross-section given in Fig. 10) is subject to a mid-span point loadapplied at the shear center. Four spans are examined in this example;2000m, 4000m, 6000m and 8000mm. The lateral torsional bucklingload estimated based on the present study are compared to those ofobtained from the classical BG element. Sectional properties are

Ixx ¼ 1:2� 109 mm4, Iyy ¼ 3:73� 108 mm4, A¼ 3� 104 mm2,Iωω ¼ 2:39� 1013 mm6, Ipx ¼ 1:70� 1010 mm5, yA ¼ �105 mm,J ¼ 4:0� 106 mm4, Dxx ¼ 16;000 mm2, Dhh ¼ 1:44� 109 mm4,Dyy ¼ 14;000 mm2, Dxh ¼ �2:107� 106 mm3, and Dyk ¼ 4:27�105 mm3.

As dicussed in previous examples, as the beam span increases,shear deformation effects become less significant. Consequently inTable 7, the buckling load ratio varies from 0.82 at a span of 2 m to0.97 at a span of 12 m.

6. Summary and conclusions

1. A general shear deformable element was developed for buck-ling analysis of members with mono-symmetric sections.

2. Compared to the shear deformable WM element [5], thenumber of degrees of freedom needed for convergence wasobserved to reduce significantly resulting.

3. A closed-form solution was derived for the buckling momentsof shear deformable mono-symmetric simply supportedbeams under uniform bending moments.

4. Results obtained based on the present element and the WMelement, were found to be in close agreement.

5. For long spans, excellent agreement was obtained with ABAQUSFEA shell results. For shorter spans, the present solution provideshigher buckling predictions compared to ABAQUS results, but

Fig. 9. Normalized interaction diagram.

Table 6Load position effect on lateral torsional buckling estimates (kN) of a cantileverbeam under a tip vertical load.

Load position

(1) Present study 55.3 56.4 68.0(2) ABAQUS 53.5 54.4 63.2Percentage difference((1)�(2))/(2)

3.3% 3.7% 7.6%

Fig. 10. Dimensions of the I-girder cross-section.

Table 7Lateral torsional buckling loads (kN) for a simply supported beam under mid-spanpoint load.

Span (m) 2 4 6 8 12

Present solution 8:984� 104 1:286� 104 4154 1921 705.1

BG element 1:098� 105 1:432� 104 4521 2062 727.6

Present solution/BGelement

0.82 0.90 0.92 0.93 0.97

A. Sahraei et al. / Thin-Walled Structures 89 (2015) 212–226 223

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lower than those based on the classical BG element. This is anatural outcome of the fact that ABAQUS shell model capturesshear deformation and distortional effects, thus providing themost flexible representation, while the present solution capturesshear deformation effects but not the distortional effect and theclassical solution captures neither effect.

6. Based on the present formulation, moment gradient factorswere developed for the mono-symmetric section investigatedin the study and were shown to depend upon the end momentratio as well as the span. Beyond certain span (3 m in thepresent problem) when �1rβr0 and the lager flange isunder compression, the moment gradient factors wereobserved to become independent of the span.

7. Interaction effects between moments and axial force as well asthe load height position effects were successfully capturedthrough the present element.

Appendix A. Matrices needed to determine stiffness matrices

This appendix provides explicit expressions for matrices form-ing stiffness matrices. In Eq. (25), the elastic stiffness matrixcomponents Kf

� �; Ksv½ �; Ks½ � and geometric stiffness components

KG½ �N ; KG½ �M ; KG½ �V ; KG½ �qy; KG½ �qz were expressed as a function ofmatrices M1½ �; M2½ �; :::; M8½ �. In order to obtain elastic and geometricstiffness matrices, first, M1½ �; M2½ �; :::; M8½ � matrices should becalculated numerically as follows

Elastic stiffness due to flexural stresses

Matrix M1½ � related to Kf� �

is given by

M1½ � ¼ EIyy

0 0 0 0 0 0 0 00 0 0 0 0 0 0

4L 6L2 a1 a2 a3 a412L3 b1 b2 b3 b4

c1;1 c1;2 c1;3 c1;4c2;2 c2;3 c2;4

Sym: c3;3 c3;4c4;4

2666666666666664

3777777777777775

ðA1Þ

in which, functions ai, bi and ci;j are defined as

ai ¼ 2φ2i eLmi �1

� �bi ¼ 6φ2imi

1m2

i

þeLmi Lmi�1ð Þm2

i

!

ci;j ¼mimj Iyyφ2iφ2jþ Iωωφ4iφ4j

�eLmi þLmj �1� �

Iyy miþmj� � ðA2a–cÞ

and φij denotes the jth element of eigenvector φi

� �.

Elastic stiffness due to Saint Venant shear stress

Matrix M2½ � related to Ksv½ � is given by

M2½ � ¼ GJ

0 0 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 0 00 0 0 0 0

d1;1 d1;2 d1;3 d1;4d2;2 d2;3 d2;4

Sym: d3;3 d3;4d4;4

26666666666666664

37777777777777775

ðA3Þ

in which, function di;j are defined as

di;j ¼φ3iφ3j

�mimj� �

eLmi þ Lmj �1� �

miþmjðA4Þ

Elastic stiffness due to shear stresses

Matrix M3½ � related to Ks½ � is given by

M3½ � ¼

0 0 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 0 0α e1 e2 e3 e4

g1 f 1;2 f 1;3 f 1;4g2 f 2;3 f 2;4

Sym: g3 f 3;4g4

2666666666666664

3777777777777775

ðA5Þ

in which, functions ei, f i;j, and gi are defined as

ei ¼ 6EIyy eLmi �1ð Þ φ2i �φ1imið Þmi

f i;j ¼G eLmi þmj �1� �miþmj

Dhh φ4iφ4jþφ3iφ4jmiþφ3jφ4imjþφ3iφ3jmimj

þDxx φ2iφ2j�φ1iφ2jmi�φ1jφ2imjþφ1iφ1jmimj

�þDxh �φ2iφ4j�φ2jφ4iþφ1iφ4jmi�φ2jφ3imiþφ1jφ4imj

�φ2iφ3jmjþφ1iφ3jmimjþφ1jφ3imimj

��

gi ¼G e2Lmi �1� �

2Dhhφ3iφ4i�2Dxxφ1iφ2iþ2Dxhφ1iφ4i�2Dxhφ2iφ3i

� �2

þGmi e2Lmi �1� �

Dxxφ1i2þ2Dxhφ1iφ3iþDhhφ3i

2� �

2

þG e2Lmi �1� �

Dxxφ2i2�2Dxhφ2iφ4iþDhhφ4i

2� �

2miðA6a–cÞ

and parameter α is

α¼ �36DhhE2Iyy

2L

G Dxh2�DhhDxx

� ðA7Þ

Geometric stiffness due to normal forces

Matrix M4½ � related to KG½ �N is given by

M4½ � ¼N

0 0 0 0 0 0 0 0L L2 L3 h1 h2 h3 h4

4L33

3L42 k1 k2 k3 k49L55 n1 n2 n3 n4

p1 o1;2 o1;3 o1;4p2 o2;3 o2;4

Sym: p3 o3;4p4

2666666666666664

3777777777777775

ðA8Þ

in which, functions hi, ki, ni, oi;j and pi are defined as

hi ¼ φ1iþφ3iyA� �

eLmi �1� �

ki ¼2 φ1iþφ3iyA� �

LmieLmi �eLmi þ1� �mi

ni ¼3 φ1iþφ3iyA� �

2eLmi þL2mi2eLmi �2LmieLmi �2

�mi

2

oi;j ¼mimj eL mi þmjð Þ�1

�Aφ1iφ1jþ Ay2Aþ Ixxþ Iyy

� �φ3iφ3jþAyA φ1iφ3jþφ1jφ3i

� �A miþmj� �

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pi ¼mi e2Lmi �1� �

φ21iþ2φ1iφ3iyAþφ2

3iy2A

� �2

þmi e2Lmi �1� �

Ixxþ Iyy� �

φ23i

� �2A

ðA9a–eÞ

Geometric stiffness due to bending moments

Matrix M5½ � related to KG½ �M is given by

M5½ � ¼M

0 0 0 0 0 0 0 00 0 0 0 0 0 0

0 0 q1 q2 q3 q40 r1 r2 r3 r4

t1 s1;2 s1;3 s1;4t2 s2;3 s2;4

Sym: t3 s3;4t4

2666666666666664

3777777777777775

ðA10Þ

in which, functions qi, ri, si;j and ti are defined as

qi ¼2φ3i eLmi �1ð Þ

mi

ri ¼ 6φ3i1

mi2þ

eLmi Lmi�1ð Þmi

2

� �6EDhhIyyφ3i e

Lmi �1� �

G Dxh2�DhhDxx

si;j ¼

eL mi þmjð Þ�1 � φ2iφ3jþφ2jφ3i

�miþ φ2iφ3jþφ2jφ3i

�mjþ Ipx

Ixxφ3iφ3jmimj

� φ1iφ3jþφ1jφ3i

�mimj�2φ3iφ3jyAmimj

0B@

1CA

miþmj� �

ti ¼Ipxφ2

3imi e2Lmi �1� �2Ixx

�φ3i e2Lmi �1

� �2φ1imi�4φ2iþ2φ3imiyA� �

2ðA11a–dÞ

Geometric stiffness due to shear forces

Matrix M6½ � related to KG½ �V is given by

M6½ � ¼ V

0 0 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 0 00 u1 u2 u3 u4

w1 v1;2 v1;3 v1;4w2 v2;3 v2;4

Sym: w3 v3;4w4

2666666666666664

3777777777777775

ðA12Þ

in which, functions ui, vi;j and wi are defined as

ui ¼ �6EDhhIyyφ3i eLmi �1ð ÞGmi Dxh

2 �DhhDxxð Þ

vi;j ¼eL mi þmjð Þ�1 �

φ2iφ3jþφ2jφ3i�φ1iφ3jmi�φ1jφ3imjþDyk

Dyyφ3iφ3j miþmj

� � �miþmj� �

wi ¼φ3i e

2Lmi �1� �

Dyyφ2i�Dyyφ1imiþDykφ3imi� �

DyymiðA13a–cÞ

Geometric stiffness due to distributed transverse load

Matrix M7½ �related to KG½ �qy is given by

M7½ � ¼ qy

0 0 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 0 00 0 0 0 0

y1 x1;2 x1;3 x1;4y2 x2;3 x2;4

Sym: y3 x3;4y4

2666666666666664

3777777777777775

ðA14Þ

in which, functions xi;j and yi are defined as

xi;j ¼�φ3iφ3j eL mi þmjð Þ �1

�yA �yqyð Þ

mi þmj

yi ¼�φ3i

2 yA�yqy �

e2Lmi �1� �

2miðA15a–bÞ

Geometric stiffness due to distributed axial load

Matrix M8½ � related to KG½ �qz is given by

M8½ � ¼ qz

0 0 0 0 0 0 0 00 0 0 z1 z2 z3 z4

0 0 a1 a2 a3 a40 b1 b2 b3 b4

d1 c1;2 c1;3 c1;4

d2 c2;3 c2;4

Sym: d3 c3;4

d4

26666666666666664

37777777777777775

ðA16Þ

in which, functions zi, ai, bi, ci and di are defined as

zi ¼ �φ3iyqz eLmi �1ð Þmi

ai ¼ �2φ3iyqz1

mi2þ

eLmi Lmi�1ð Þmi

2

bi ¼ 3φ3iyqz2mi

3�eLmi L2mi

2�2Lmiþ2 �

mi3

0@

1Aþ6EDhhIyyφ3iyqz eLmi �1

� �Gmi Dxh

2�DhhDxx

ci;j ¼�yqz eL mi þmjð Þ�1

�φ2iφ3jþφ2jφ3i

�miþmj

di ¼�φ2iφ3iyqz e2Lmi �1

� �mi

ðA17a–dÞ

Load position matrix for concentrated transverse load

When a member is subject to a concentrated transverse load Qy

applied at z¼ zQy and position yQy relative to the shear center, theload function in Eq. (6g) can be demonstrated as qy zð Þ ¼QyDirac z�zQy

� �. Substituting this load function into Eq. (6g), one

can obtain a new geometric stiffness matrix KG½ �Qy due to loadposition effect relative to the shear center SC. Matrix M9½ � relatedto KG½ �Qy is given by

M9½ � ¼ �Qy yQy�yA �

0 0 0 0 0 0 0 00 0 0 0 0 0 0

0 0 0 0 0 00 0 0 0 0

f 1 e1;2 e1;3 e1;4

f 2 e2;3 e2;4

Sym: f 3 e3;4

f 4

26666666666666664

37777777777777775ðA18Þ

in which, functions ei;j and f i are defined as

ei;j ¼φ3iφ3jezQy mi þmjð Þ

f i ¼φ3i2e2zQymi ðA19a–bÞ

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